I stand corrected. I was misled by the fact that the Cornell site says the papaya is also called a pawpaw.

Here's a page from Virginia Tech that has some good pictures of Asimina triloba.

I think I recall my mother (now age 90) saying that she was familiar with pawpaws where she grew up in Arkansas (We don't have them where I live in Minnesota) and that her mother liked to eat them, but nobody else in the family liked them. Mind you, poor people in that region, at that time, ate a lot of things we wouldn't eat today, because they needed the nourishment—possums, for instance.

This North Carolina State University site on poisonous plants says that pawpaws are "edible but some people suffer severe stomach and intestinal pain." It also says "The fruits generally fall to the ground before they are ripe and therefore must be ripened at home. Ripen outdoors -- the smell is overpowering."

This gives more meaning to the line "Pickin' up pawpaws; put 'em in your basket [or pocket]" – you pick them up; you don't pick them off the tree – and you put them in your basket or pocket; you don't eat them directly. The song contains advice on what to do with pawpaws!